The Ninth British Mathematical Olympiad
1973年第九届英国奥林匹克数学竞赛 
 A variable circle touches two fixed circles at P and Q. Show that the line PQ passes through one of two fixed points. State a generalisation to ellipses or conics.
 Given any nine points in the interior of a unit square, show that we can choose 3 which form a triangle of area at most 1/8.
 The curve C is the quarter circle x^{2} + y^{2} = r^{2}, x ≥ 0, y ≥ 0 and the line segment x = r, 0 ≥y ≥ h. C is rotated about the yaxis for form a surface of revolution which is a hemisphere capping a cylinder. An elastic string is stretched over the surface between (x, y, z) = (r sin θ, r cos θ, 0) and (r, h, 0). Show that if tan θ > r/h, then the string does not lie in the xy plane. You may assume spherical triangle formulae such as cos a = cos b cos c + sin b sin c cos A, or sin A cot B = sin c cot b  cos c cos A.
 n equilateral triangles side 1 can be fitted together to form a convex equiangular hexagon. The three smallest possible values of n are 6, 10 or 13. Find all possible n .
 Show that there is an infinite set of positive integers of the form 2^{n}  7 no two of which have a common factor .
 The probability that a teacher will answer a random question correctly is p. The probability that randomly chosen boy in the class will answer correctly is q and the probability that a randomly chosen girl in the class will answer correctly is r. The probability that a randomly chosen pupil's answer is the same as the teacher's answer is 1/2. Find the proportion of boys in the class .
 From each 10000 live births, tables show that y will still be alive x years later. y(60) = 4820 and y(80) = 3205, and for some A, B the curve Ax(100x) + B/(x40)^{2} fits the data well for 60 <= x <= 100. Anyone still alive at 100 is killed. Find the life expectancy in years to the nearest 0.1 year of someone aged 70 .
 T: z → (az + b)/(cz + d) is a map. M is the associated matrix
a b
c d
Show that if M is associated with T and M' with T' then the matrix MM' is associated with the map TT'. Find conditions on a, b, c, d for T^{4} to be the identity map, but T^{2} not to be the identity map.
 Let L(θ) be the determinant:
x y 1
a + c cos θ b + c sin θ 1
l + n cos θ m + n sin θ 1
Show that the lines are concurrent and find their point of intersection.
 Write a computer program to print out all positive integers up to 100 of the
form a^{2}  b^{2}  c^{2} where a, b, c are positive integers and a ≥ b + c .
 (1) A uniform rough cylinder with radius a, mass M, moment of inertia Ma^{2}/2 about its axis, lies on a rough horizontal table. Another rough cylinder radius b, mass m, moment of inertia mb^{2}/2 about its axis, rests on top of the first with its axis parallel. The cylinders start to roll. The plane containing the axes makes the angle θ with the vertical. Show the forces during the period when there is no slipping. Write down equations, which will give on elimination a differential equation for q, but you do not need to find the differential equation.
(2) Such a differential equation is θ_{2}(4 + 2 cos θ  2 cos^{2}θ + 9k/2) + θ_{1}^{2} sin θ (2 cos θ  1) = 3g(1 + k) (sin θ /(a + b), where k = M/m. Find θ_{1} in terms of θ. Here θ_{1} denotes dθ/dt and θ_{2} denotes the second derivative.

点击此处查看相关视频讲解 

